In mathematics, the Heilbronn triangle problem is a typical question in the area of irregularities of distribution , within elementary geometry.
Consider region D in the plane: a unit circle or general polygon — the asymptotics of the problem, which are the interesting aspect, aren't dependent on the exact shape. Place a number n of distinct points (greater than three) within D: there are numerous triangles that can be constructed from three of those points. For a given configuration, we are interested knowing that there is some small triangle, and how small. The Heilbronn triangle problem involves therefore the extremal case: minimum area from these points. The question was posed by Hans Heilbronn , of giving a lower bound for this minimum area, denoted by Δ(n). This is therefore formally of the shape
- Δ(n) = maxX in C(D,n) mintriangles T of X Area of T.
The notation here says that X is a configuration of n points in D, and T is a triangle with three points of X as vertices.
Heilbronn initially conjectured that this area would be of order
- Δ(n) ~ Constant·n−2;
however there have been improvements to this bound.
As of 2004 it is known that
- A·n−2log n ≤ Δ(n) ≤ B·n−8/7exp(C√log n)
where A, B and C are constants.
References
- Komlos, J.; Pintz, J.; and Szemerédi, E. On Heilbronn's Triangle Problem. J. London Math. Soc. 24, 385-396, 1981.
- Komlos, J.; Pintz, J.; and Szemerédi, E. A Lower Bound for Heilbronn's Triangle Problem. J. London Math. Soc. 25, 13-24, 1982.
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